1. Field of the Invention
The invention relates to wave mode converters and, more particularly, to wave non-confining mode to wave confining mode converters. The invention also concerns laser devices transmitting confining mode waves.
When a laser beam is focused on to a material, a plasma is formed. It has been found that the propagation mode of the wave focused on to the material has a very important effect on the formation of the plasma and the confinement thereof.
The object of the invention is to control and confine a plasma jet by (a) forcing the laser beam to propagate in particular modes to be described hereinafter and (b) subjecting the plasma to a magnetic field having a steep gradient in the direction of the axis of propagation. Generally, the confining modes are wave beam modes wherein the electric field vector at a point in a wave plane has a time phase shift equal or proportional to the azimut of the point considered with respect to the beam axis. Further the confining mode waves are circularly polarized waves. The advantage of these forms of mode propagation is that they constitute the most effective method of confining the plasma along the beam propagation axis.
In official wave nomenclature, the wave mode if followed by two subscripts. The first indicates the number of space periods in the azimutal direction, and the second indicates the number of space petiods in the radial direction. In the present specification we shall never write the second subscript, since from this point of view, the mode will always be close to the fundamental conditions, with a relatively small proportion of harmonics which may vary in dependence on the distance from the beam axis when the wave amplitude is varied.
The magnetic field can be used to obtain resonance, in the neighbourhood of which the confining forces are considerably increased. Furthermore, when the axial magnetic field has a steep axial gradient, the observed phenomena changes with the sign of the gradient. If the field maximum is on the side of the material receiving the impact, the ions rotating in the plasma rings are compressed against the material, which increases impacts between ions and assists approach of nuclear fusion phenomena. If, on the other hand, the mininum magnetic field is on the side of the material, the ions rotating in the rings move away from the material, the plasma is stretched along the axis, and the ions rotating in the rings are regrouped at a certain distance from the material, which assists the emission of coherent X-rays.
We shall now describe the configuration of waves having circular polarization and a positive or negative azimuth phase-shift.
In a phase plane referred to by the axes Ox, Oy and taken as complex reference plane, the electric field of a circular-polarized plane wave having the angular frequency .omega. can be represented by the complex number: EQU E=E.sub.o exp(j.omega.t)
where E.sub.o is a reference amplitude of the electric field and j=.sqroot.-1. Such a wave is said to be azimuthally phase-shifted around the propagation axis, taken as the Oz axis, if, with respect to the previously-defined plane wave, it undergoes a phase shift or rotation of the polarisation vector proportional to the azimuth .phi.. The phase shift must comprise a whole number of periods per complete aziputh revolution around the axis, but the electric field amplitude cannot be uniform since it must be zero along the axis where the azimuth is indeterminate. In addition the wave must be represented by an analytic function. Consequently, near the axis, at a short distance r therefrom, the electric field vector is represented by the following complex number: EQU E=E.sub.o (r/r.sub.o).sup.N exp j (.omega.t.+-.N.phi.) (1)
where N is an integer and r.sub.o is a reference radius vector.
The preceding is the first term in the series expansion of a function which, when r increases, passes through a maximum and subsequently falls to zero.
These waves vary greatly depending whether the phase-shift varies in the direction of .omega.t (in which case it is said to be positive) or in the opposite direction (when it is called negative).
In the case of a positively phase-shifted first-order wave (for which N=1), the electric field is represented by the complex number: ##EQU1## and can be considered as the sum of two fields of vectors corresponding respectively to the following complex numbers: ##EQU2##
Expression (2a) denotes a field of radially-disposed electric vectors, like those of the TM.sub.O mode in circular waveguides, the lines of force being radii indicated by chain lines in FIG. 1. Expression (2b) denotes a field of orthoradial vectors having lines of force denoted by circles indicated by continuous lines in FIG. 1. The circular lines of force are similar to those of the TE.sub.O mode in a circular waveguide (as already said, the single subscript denotes the number of aziputhal space periods).
The field of electric vectors of a first-order aziputhally phase-shifted circular-polarized wave, the phase-shift being negative, can be represented by the complex number: ##EQU3## i.e. the sum of the fields: ##EQU4##
The fields of vectors corresponding to expressions (3a) and (3b) are very similar to fields of electric vectors of two orthogonal waves in phase quadrature denoted by TE.sub.2 in the theory of circular waveguides.
The lines of force can be determined as follows. At a given point P (FIG. 2), the electroc field is at an angle -2.phi. to the radius vector OP. It is known that, in the case of two points P and P' very close together and having the coordinates r, .phi. and r+dr, .phi.+.phi. respectively, the angle between the direction PP' and the radius vector has the tangent rd.phi./dr. Consequently, the lines of forces are governed by the following differential equation: EQU rd.phi./dr=-tg(2.phi.) (4)
which is integrated to: ##EQU5## k being a constant. In Cartesian coordinates, expression (4) is written: EQU r.sup.2 sin .phi. cos .phi.=xy=k.sup.2 /2 (4')
The lines of forces are equilateral hyperbolas, as shown in continuous lines in FIG. 2. If the azimuth origin is shifted by .pi./4, we obtain a second family of lines of force, consisting of equilateral hyperbolas orthogonal to the preceding ones and shown by chain lines in FIG. 2. The two fields of vectors vibrate in phase quadrature. At any point on the plane, the wave is circular-polarized. There is no privileged point. For this reason, point M on the Ox axis (FIG. 2) at a distance r.sub.o from the origin can be considered as a current point. A rectilinear radial line of force exciting a cylindrical vibration in the plasma passes through point M, and so does a hyperbolic orthoradial line of force having a radius of curvature equal to the distance from the origin (the radius of curvature of an equilateral hyperbola at its apex is equal to the distance between the apex and the center of symmetry of the equilateral hyperbola). When a charged particle moves under the action of an electric field having curved lines of force, the particle is subjected to an electric centrifugal force which tends to move it away from the center of curvature. In the case in FIG. 2, the lines of force have their convexity facing the beam axis and the particle approaches the axis on moving away from the center of curvature. In the case of the wave expressed by (2b) and shown in FIG. 1, the lines of force have their convexity facing the exterior, i.e. the electric centrifugal force moves the particle away from the axis. In this case, however, as we shall see, allowance must be made for a magnetic confining force, which is not negligible as in FIG. 2.
Before discussing the confining forces in greater detail, we shall study certain properties of circular-polarized waves having a positive or negative azimuthal phase shift, in the general case of the N.sup.th order. These properties are general versions of those found for the waves in equations (2) and (3) in the case of the first order (N=1).
By introducing the complex number .xi. EQU .xi.=r exp j.phi. (5)
which denotes a point M on the complex plane of radius vector r and azimuth .phi., the electric field of circular-polarized waves having N.sup.th order azimuthal phase-shift can be rewritten in the form of a complex number: ##EQU6## for a positive phase-shift and ##EQU7## for a negative phase shift, where .xi.* denotes the conjugate complex of .xi..
The vector fields E defined by (6) and (7) are general versions of expressions (2) and (3). At a given point on the phase plane, E is a rotating vector which can be broken up into two sinusoidal vectors in fixed directions, perpendicular to one another and represented by the numbers E' and E": ##EQU8##
If we put the positive sign in front of jN for expressions derived from (6) and the negative sign for expressions from (7), the vectors rotate by .+-.N.phi. when there is an increase of .phi. in the azimuth of the point where they are considered.
In calculating the lines of force of vectors E' or E", we shall start from the fact that the tangent to these curves is at an angle (.+-.N-1).phi. to the radius vector.
The differential expression for the tangent to the line of force gives the differential equation of these lines ##EQU9##
By integration, we obtain the general equation of the lines of forces: ##EQU10##
By way of example, in the case of positively phase-shifted waves, we must put the positive sign in front of N and (N-1) is zero for the first order. In that case, the differential equation is indefinite and its solutions are circles r-r.sub.o and radii .phi.=.phi..sub.o characteristic of the modes TE.sub.O and TM.sub.O respectively and shown in FIG. 1.
An important case is where N is zero. Then equation (10) is as follows, after rearrangement: EQU r sin (.phi.-.phi..sub.o)=r.sub.o
This is a family of parallel straight lines. The corresponding wave is a rectilinear-polarized plane wave.
In the case of waves having first-order negative azimuthal phase-shoft, we have N=1 and a minus sign in front of N. Equation (10) is written as follows: EQU r=r.sub.o [sin 2(.phi.-.phi..sub.o)].sup.-1/2
If the two members are squared and we change over to Cartesian coordinates, we can immediately see that the lines of force are hyperbolas: EQU r.sup.2 .times.2 sin (.phi.-.phi..sub.o) cos (.phi.-.phi..sub.o)=2xy=r.sub.o.sup.2
as shown in FIG. 2.
The following is of importance for mode converters using half-wave plates and described hereinafter. Given two fields of vectors defined by complex numbers such as E' or E" in expressions (8), one corresponding to a value .+-.N.sub.1 and the other to a value .+-.N.sub.2, we can define the curved which is at any point tangent to the bisector of the two vectors. The tangent to this curve makes the following angle to the radius vector: ##EQU11##
Accordingly, the curve can be represented by an equation such as (10) in which N is equal to the arithmetic mean of N.sub.1 and N.sub.2.
In the present application, we shall show how a non-confining mode can be converted into a confining mode using contiguous sector-shaped half-wave plates, the fast axes of the sectors being approximately tangential to curves which, at all points bisect the curves of the field to be converted from one mode to the other. The calculation given hereinbefore is a model for the general calculation of these bisecting curves.
Clearly, field expressions such as (6) and (7) can be valid only over a restricted range, since the modulus increases indefinitely with distance from the beam axis.
This is unimportant when waveguides are used in the centimeter wave region, since in that case the range is accurately delimited. However, in the infrared or light wave region, it is necessary to focus in free space. Whereas in the previous case the electric-flux displacement current was closed by a conduction current flowing in the waveguide walls, it must now close in free space. Outside the range where expressions (6) and (7) remain valid, a "closure field" must be added to the "confining wave" field represented by (6) and (7). The set of fields can be broken up into the first term of a series expansion expressing these modes. We shall not discuss this breakdown in detail.